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Two-site entanglement in the two-dimensional Hubbard model

Frederic Bippus1, Anna Kauch1, Gergő Roósz2, Christian Mayrhofer1, Fakher Assaad3,4, and Karsten Held1

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Phys. Rev. B 113, 035152 – Published 26 January, 2026

DOI: https://doi.org/10.1103/81gy-zz9l

Abstract

The study of entanglement in strongly correlated electron systems typically requires knowledge of the reduced density matrix. Here, we apply the parquet dynamical vertex approximation to study the two-site reduced density matrix at varying distance, in the Hubbard model at weak coupling. This allows us to investigate the spatial structure of entanglement in dependence of interaction strength, electron filling, and temperature. We compare results from different entanglement measures, and benchmark against quantum Monte Carlo.

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References (96)

  1. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  2. C. J. Halboth and W. Metzner, d-wave superconductivity and Pomeranchuk instability in the two-dimensional Hubbard model, Phys. Rev. Lett. 85, 5162 (2000).
  3. C. Honerkamp, M. Salmhofer, N. Furukawa, and T. M. Rice, Breakdown of the Landau-Fermi liquid in two dimensions due to umklapp scattering, Phys. Rev. B 63, 035109 (2001).
  4. E. Gull and A. Millis, Numerical models come of age, Nat. Phys. 11, 808 (2015).
  5. M. Kitatani, L. Si, O. Janson, R. Arita, Z. Zhong, and K. Held, Nickelate superconductors – a renaissance of the one-band Hubbard model, npj Quantum Mater. 5, 59 (2020).
  6. S. Sakai, Y. Motome, and M. Imada, Evolution of electronic structure of doped mott insulators: Reconstruction of poles and zeros of Green's function, Phys. Rev. Lett. 102, 056404 (2009).
  7. G. Sordi, P. Sémon, K. Haule, and A.-M. S. Tremblay, Strong coupling superconductivity, pseudogap, and Mott transition, Phys. Rev. Lett. 108, 216401 (2012).
  8. F. Šimkovic, R. Rossi, A. Georges, and M. Ferrero, Origin and fate of the pseudogap in the doped Hubbard model, Science 385, eade9194 (2024).
  9. K. Held, R. Peters, and A. Toschi, Poor man's understanding of kinks originating from strong electronic correlations, Phys. Rev. Lett. 110, 246402 (2013).
  10. K. Byczuk, M. Kollar, K. Held, Y.-F. Yang, I. A. Nekrasov, T. Pruschke, and D. Vollhardt, Kinks in the dispersion of strongly correlated electrons, Nat. Phys. 3, 168 (2007).
  11. J. Krsnik and K. Held, Local correlations necessitate waterfalls as a connection between quasiparticle band and developing hubbard bands, Nat. Commun. 16, 255 (2025).
  12. F. Gebhard, The Mott Metal-Insulator Transition (Springer, Berlin, 1997).
  13. A. Georges and G. Kotliar, Hubbard model in infinite dimensions, Phys. Rev. B 45, 6479 (1992).
  14. P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012).
  15. P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Measuring multipartite entanglement through dynamic susceptibilities, Nat. Phys. 12, 778 (2016).
  16. F. Mazza, S. Biswas, X. Yan, A. Prokofiev, P. Steffens, Q. Si, F. F. Assaad, and S. Paschen, Quantum fisher information in a strange metal, arXiv:2403.12779.
  17. P. Laurell, A. Scheie, E. Dagotto, and D. A. Tennant, Witnessing entanglement and quantum correlations in condensed matter: A review, Adv. Quantum Technol. 8, 2400196 (2024).
  18. I. Frérot and T. Roscilde, Quantum variance: A measure of quantum coherence and quantum correlations for many-body systems, Phys. Rev. B 94, 075121 (2016).
  19. F. Bippus, J. Krsnik, M. Kitatani, L. Akšamović, A. Kauch, N. Barišić, and K. Held, Entanglement in the pseudogap regime of cuprate superconductors, Phys. Rev. B 112, L081110 (2025).
  20. D. Bałut, X. Guo, N. de Vries, D. Chaudhuri, B. Bradlyn, P. Abbamonte, and P. W. Phillips, Quantum fisher information reveals UV-IR mixing in the strange metal, arXiv:2412.14413.
  21. G. Bellomia, Quantum information insights into strongly correlated electrons, Ph.D. thesis, SISSA, 2024.
  22. S. Bera, A. Haldar, and S. Banerjee, Dynamical mean-field theory for rényi entanglement entropy and mutual information in the hubbard model, Phys. Rev. B 109, 035156 (2024).
  23. G. Bellomia, C. Mejuto-Zaera, M. Capone, and A. Amaricci, Quasilocal entanglement across the mott-hubbard transition, Phys. Rev. B 109, 115104 (2024).
  24. C. Walsh, P. Sémon, D. Poulin, G. Sordi, and A.-M. S. Tremblay, Local entanglement entropy and mutual information across the Mott transition in the two-dimensional Hubbard model, Phys. Rev. Lett. 122, 067203 (2019).
  25. C. Walsh, P. Sémon, D. Poulin, G. Sordi, and A.-M. S. Tremblay, Entanglement and classical correlations at the doping-driven Mott transition in the two-dimensional Hubbard model, PRX Quantum 1, 020310 (2020).
  26. A. Majumder and S. Sen, Mott metal-insulator transition in a modified periodic Anderson model: Insights from entanglement entropy and role of short-range spatial correlations, Adv. Theory Simul. 9, e01224 (2025).
  27. G. Roósz, A. Kauch, F. Bippus, D. Wieser, and K. Held, Two-site reduced density matrix from one- and two-particle Green's functions, Phys. Rev. B 110, 075115 (2024).
  28. S. Rohshap, H. Ishida, F. Bippus, A. Kauch, K. Held, H. Shinaoka, and M. Wallerberger, Diagnosing phase transitions through time scale entanglement, arXiv:2507.11276.
  29. C. Gauvin-Ndiaye, J. Tindall, J. R. Moreno, and A. Georges, Mott transition and volume law entanglement with neural quantum states, Phys. Rev. Lett. 134, 076502 (2025).
  30. G. Ehlers, J. Sólyom, Ö. Legeza, and R. M. Noack, Entanglement structure of the Hubbard model in momentum space, Phys. Rev. B 92, 235116 (2015).
  31. S.-J. Gu, S.-S. Deng, Y.-Q. Li, and H.-Q. Lin, Entanglement and quantum phase transition in the extended Hubbard model, Phys. Rev. Lett. 93, 086402 (2004).
  32. S. Abaach, Z. Mzaouali, and M. El Baz, Long distance entanglement and high-dimensional quantum teleportation in the Fermi–Hubbard model, Sci. Rep. 13, 964 (2023).
  33. M. Lo Schiavo, F. Cipriani, G. De Riso, A. Romano, and C. Noce, Quantum entanglement in an extended Hubbard model as evaluated from a spin concurrence measure, J. Magn. Magn. Mater. 584, 171066 (2023).
  34. F.-H. Wang and X. Y. Xu, Entanglement Rényi negativity of interacting fermions from quantum Monte Carlo simulations, Nat. Commun. 16, 2637 (2025).
  35. M. Consiglio, J. Odavić, R. Bonsignori, G. Torre, M. Wieśniak, F. Franchini, S. M. Giampaolo, and T. J. G. Apollaro, Long-distance genuine multipartite entanglement between magnetic defects in spin chains, Phys. Rev. A 111, 032434 (2025).
  36. J. D'Emidio, R. Orús, N. Laflorencie, and F. de Juan, Universal features of entanglement entropy in the honeycomb Hubbard model, Phys. Rev. Lett. 132, 076502 (2024).
  37. O. Vafek, N. Regnault, and B. A. Bernevig, Entanglement of exact excited eigenstates of the Hubbard model in arbitrary dimension, SciPost Phys. 3, 043 (2017).
  38. G. De Chiara and A. Sanpera, Genuine quantum correlations in quantum many-body systems: A review of recent progress, Rep. Prog. Phys. 81, 074002 (2018).
  39. S. Yamashika, S. Endo, and H. Tajima, Quantum Fisher information as a measure of symmetry breaking in quantum many-body systems, arXiv:2509.07468.
  40. G. Barcza, O. Legeza, K. H. Marti, and M. Reiher, Quantum-information analysis of electronic states of different molecular structures, Phys. Rev. A 83, 012508 (2011).
  41. G. D. Chiara, C. Brukner, R. Fazio, G. M. Palma, and V. Vedral, A scheme for entanglement extraction from a solid, New J. Phys. 8, 95 (2006).
  42. A. Toschi, A. A. Katanin, and K. Held, Dynamical vertex approximation: A step beyond dynamical mean-field theory, Phys. Rev. B 75, 045118 (2007).
  43. A. Valli, T. Schäfer, P. Thunström, G. Rohringer, S. Andergassen, G. Sangiovanni, K. Held, and A. Toschi, Dynamical vertex approximation in its parquet implementation: Application to Hubbard nanorings, Phys. Rev. B 91, 115115 (2015).
  44. G. Rohringer, H. Hafermann, A. Toschi, A. A. Katanin, A. E. Antipov, M. I. Katsnelson, A. I. Lichtenstein, A. N. Rubtsov, and K. Held, Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory, Rev. Mod. Phys. 90, 025003 (2018).
  45. W. H. Zurek, Information Transfer in Quantum Measurements: Irreversibility and Amplification (Springer, Boston, 1983).
  46. S. M. Barnett and S. J. D. Phoenix, Entropy as a measure of quantum optical correlation, Phys. Rev. A 40, 2404 (1989).
  47. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  48. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  49. H. Shapourian and S. Ryu, Entanglement negativity of fermions: Monotonicity, separability criterion, and classification of few-mode states, Phys. Rev. A 99, 022310 (2019).
  50. H. Shapourian, K. Shiozaki, and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev. B 95, 165101 (2017).
  51. R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Monte Carlo calculations of coupled boson-fermion systems, Phys. Rev. D 24, 2278 (1981).
  52. S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gubernatis, and R. T. Scalettar, Numerical study of the two-dimensional Hubbard model, Phys. Rev. B 40, 506 (1989).
  53. F. Assaad and H. Evertz, World-line and determinantal quantum Monte Carlo methods for spins, phonons and electrons, in Computational Many-Particle Physics, Lecture Notes in Physics Vol. 739, edited by H. Fehske, R. Schneider, and A. Weiße (Springer, Berlin, 2008), pp. 277–356.
  54. F. F. Assaad, M. Bercx, F. Goth, A. Götz, J. Hofmann, E. Huffman, Z. Liu, F. Parisen Toldin, J. Portela, and J. Schwab, The ALF (algorithms for lattice fermions) project release 2.0. Documentation for the auxiliary-field quantum Monte Carlo code, SciPost Phys. Codebases 1 (2022).
  55. T. Grover, Entanglement of interacting fermions in quantum Monte Carlo calculations, Phys. Rev. Lett. 111, 130402 (2013).
  56. F. F. Assaad, T. C. Lang, and F. Parisen Toldin, Entanglement spectra of interacting fermions in quantum Monte Carlo simulations, Phys. Rev. B 89, 125121 (2014).
  57. In [27], all elements are expressed in terms of second-quantized operators. Here, we want to point a typo therin. In Eq. (34) the hopping spins should be flipped: A9,6=−12(ci↓†cj↓+cj↓†ci↓)(1−ni↑)nj,↑.
  58. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  59. L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80, 517 (2008).
  60. N. Friis, G. Vitagliano, M. Malik, and M. Huber, Entanglement certification from theory to experiment, Nat. Rev. Phys. 1, 72 (2019).
  61. A. Rényi, On measures of entropy and information, Berkeley Symposium on Mathematical Statistics and Probability, edited by J. Neyman (Scientific Research, Berkeley, 1961), pp. 547–561.
  62. The Rényi mutual information is based on the generalized Rényi entropy Sα=ln(trρα)/(1−α). Our definition of the Rényi mutual information Eq. (5) corresponds to α=2. For α=1 the original von Neumann entropy and its mutual information Eq. (4) is recovered.
  63. M. Kormos and Z. Zimborás, Temperature driven quenches in the Ising model: Appearance of negative Rényi mutual information, J. Phys. A: Math. Theor. 50, 264005 (2017).
  64. J. Kudler-Flam, Rényi mutual information in quantum field theory, Phys. Rev. Lett. 130, 021603 (2023).
  65. In the literature, the negativity is also often considered as an upper bound for entanglement. In these cases, its property to bound distillable entanglement from above is referenced [96].
  66. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  67. M. B. Plenio, Logarithmic negativity: A full entanglement monotone that is not convex, Phys. Rev. Lett. 95, 090503 (2005).
  68. J. Eisert, V. Eisler, and Z. Zimborás, Entanglement negativity bounds for fermionic Gaussian states, Phys. Rev. B 97, 165123 (2018).
  69. A. D. Gottlieb and N. J. Mauser, Properties of nonfreeness: An entropy measure of electron correlation, Int. J. Quantum Inf. 05, 815 (2007).
  70. K. Held and N. J. Mauser, Physics behind the minimum of relative entropy measures for correlations, Eur. Phys. J. B 86, 328 (2013).
  71. G. Bellomia, A. Amaricci, and M. Capone, Local classical correlations between physical electrons in the Hubbard model, arXiv:2506.18709.
  72. M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull, The Hubbard model: A computational perspective, Annu. Rev. Condens. Matter Phys. 13, 275 (2022).
  73. H. Tasaki, The Hubbard model - an introduction and selected rigorous results, J. Phys.: Condens. Matter 10, 4353 (1998).
  74. T. Schäfer, N. Wentzell, F. Šimkovic, Y.-Y. He, C. Hille, M. Klett, C. J. Eckhardt, B. Arzhang, V. Harkov, F.-M. Le Régent, A. Kirsch, Y. Wang, A. J. Kim, E. Kozik, E. A. Stepanov, A. Kauch, S. Andergassen, P. Hansmann, D. Rohe, Y. M. Vilk, et al., Tracking the footprints of spin fluctuations: A multimethod, multimessenger study of the two-dimensional Hubbard model, Phys. Rev. X 11, 011058 (2021).
  75. F. Krien, A. Kauch, and K. Held, Tiling with triangles: Parquet and GWγ methods unified, Phys. Rev. Res. 3, 013149 (2021).
  76. S. Iskakov, H. Terletska, and E. Gull, Single- and two-particle finite size effects in interacting lattice systems, Phys. Rev. B 106, 235106 (2022).
  77. M. Wallerberger, A. Hausoel, P. Gunacker, A. Kowalski, N. Parragh, F. Goth, K. Held, and G. Sangiovanni, W2dynamics: Local one- and two-particle quantities from dynamical mean field theory, Comput. Phys. Commun. 235, 388 (2019).
  78. C. De Dominicis and P. C. Martin, Stationary entropy principle and renormalization in normal and superfluid systems. I. Algebraic formulation, J. Math. Phys. 5, 14 (1964).
  79. C. De Dominicis and P. C. Martin, Stationary entropy principle and renormalization in normal and superfluid systems. II. Diagrammatic formulation, J. Math. Phys. 5, 31 (1964).
  80. N. E. Bickers, Self-consistent many-body theory for condensed matter systems, in Theoretical Methods for Strongly Correlated Electrons, edited by D. Sénéchal, A.-M. Tremblay, and C. Bourbonnais (Springer, New York, 2004), pp. 237–296.
  81. F. Krien and A. Kauch, The plain and simple parquet approximation: Single-and multi-boson exchange in the two-dimensional Hubbard model, Eur. Phys. J B 95, 69 (2022).
  82. S. Rohshap, M. K. Ritter, H. Shinaoka, J. von Delft, M. Wallerberger, and A. Kauch, Two-particle calculations with quantics tensor trains: Solving the parquet equations, Phys. Rev. Res. 7, 023087 (2025).
  83. F. Krien, A. Valli, and M. Capone, Single-boson exchange decomposition of the vertex function, Phys. Rev. B 100, 155149 (2019).
  84. Note that only the symmetry-reduced wedge (Δx,Δy),Δx>0,Δx≥Δy has been directly computed in pDΓA and the remaining space has been filled by symmetry relations.
  85. C. Hille, F. B. Kugler, C. J. Eckhardt, Y.-Y. He, A. Kauch, C. Honerkamp, A. Toschi, and S. Andergassen, Quantitative functional renormalization group description of the two-dimensional Hubbard model, Phys. Rev. Res. 2, 033372 (2020).
  86. D. Wieser, Single-boson exchange decomposition for the extended Hubbard model with application to the optical conductivity, Master thesis, TU Wien, 2025.
  87. H. Q. Lin and J. E. Hirsch, Two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping, Phys. Rev. B 35, 3359 (1987).
  88. J. A. Smolin, J. M. Gambetta, and G. Smith, Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise, Phys. Rev. Lett. 108, 070502 (2012).
  89. M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100, 070502 (2008).
  90. A. Chincholi, S. Capponi, and F. Alet, Detecting the largest correlations using the correlation density matrix: A quantum monte carlo approach, arXiv:2507.21697.
  91. J.-M. Lihm, D. Kiese, S.-S. B. Lee, and F. B. Kugler, The finite-difference parquet method: Enhanced electron-paramagnon scattering opens a pseudogap, arXiv:2505.20116.
  92. F. Bippus, A. K. Kauch, G. Roósz, C. Mayrhofer, F. Assaad, and K. Held, Two-site entanglement in the two-dimensional Hubbard model, https://doi.org/10.48436/7pft0-nn697.
  93. S. M. Girvin and K. Yang, Modern Condensed Matter Physics (Cambridge University Press, Cambridge, 2019).
  94. Y. Ben-Aryeh and A. Mann, Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems, Int. J. Quantum Inf. 13, 1550061 (2015).
  95. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A: Math. Theor. 42, 504003 (2009).
  96. X. Wang and M. M. Wilde, α-logarithmic negativity, Phys. Rev. A 102, 032416 (2020).

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Phys. Rev. B 113, 035152– Published 26 January, 2026
  • Received 11 June 2025
  • Revised 16 December 2025
  • Accepted 22 December 2025
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